Solved Vector Questions

Vector topic in physics. Addition, subtraction, division and multiplication operations on vectors. Separation of vectors into its components. Solved questions.

Question 1

Three vector (K, L and M) is seen in the figure above.

What is magnitude and direction of 2K – 2L + M operation?

Solution:

If we want to multiply a vector by a scalar number, we multiply magnitude of the vector by the scalar number.

Magnitude of the vector will be equal the result of the multiply.

If sign of the scalar number that we multiply by the vector is negative, the direction of the vector changes. It is opposite.

If sign of the scalar number that we multiply by the vector is positive, the direction of the vector not changes.

Added of the vectors is easy on the unit square surface. Head of one of vectors is placed to the tail of the other. All vectors are added in this way. The vector drawn to the head of the last vector from tail of the first vector is the resultant vector.

It is shown how to does the 2K – 2L + M operation in the following.

The magnitude of the vector can find using Pythagorean theorem. The resultant vector is 5 units long and its direction is direction Northeast.

Question -2

In the figure above, the lenght of each unit square is equal to 50 N.

What is the magnitude and the direction of resultant vector of the M – 2K + L – (N/2) operation?

Solution:

We won’t any change on the vector M and the vector L.

We are reverse the vector K and we are multiplied by 2.

We will divide the vector N by 2 and we will reverse it.

If we add these vectors by head – to – tail method, we obtain the resultant vector.

The length of the vector is 7 unit squares. Its magnitude is,
R = 7.50 = 350 N
Direction of the vector R is eastward.

Question 3

It is seen four vectors in the above figure.

Magnitude of the vector K is 50 N, magnitude of the L is 30 N, magnitude of the M is 20 N, and magnitude of the N is 50 N.

Find the resultant of the K – (3L/2) – 3M + 2N.

(sin70° = 0,94, cos70° = 0,34, sin45° = cos45° = 0,7)

Solution:

First, we must multiply the vector "K" by 1, multiply the vector "L" by (-3/2), multiply the vector "M" by (-3) and multiply the vector "N" by 2.

K = 50 N

-3L/2 = - 45N

-3M = -60 N

N = 50 N

Later, we separates each of these vectors into its components.
Finally, we add these components and we find the resultant vector.

The components of the vector K.

Kx = K.cos70°

Kx = 50.0,34

Kx = 17 N

Ky = K.sin70°

Ky = 50.0,94

Ky = 47 N

The components of the vector L.

Lx = L.cos90°

Lx = 45.0

Lx = 0

Ly = L.sin90°

Ly = 45.1

Ly = -45 N

The components of the vector M.

Mx = M.cos90°

Mx = 60.0

Mx = 0

My = M.sin90°

My = 60.1

My = 60 N

The components of the vector N.

Nx = N.cos45°

Nx = 100.0,7

Nx = 70 N

Ny = N.sin45°

Ny = 100.0,7

Ny = 70 N

Now, let we add the components that on the x and y axis among themselves.

Rx = Kx + Lx + Mx + Nx

Rx = 17 + 0 + 0 + 70

Rx = 87 N

Ry = Ky + Ly + My + Ny

Ry = 47 – 45 + 60 + 70

Ry = 132 N

Now, we have two vectors perpendicular to each other. These vectors are components of the resultant vector. The vector R can found using components.

From the Pythagorean theorem,

(Ry)2 + (Rx)2 = R2

R2 = 1322 + 872

R2 = 7569 + 17424

R2 = 24993

R = 158 N

Magnitude of the vector R is 158 N and its direction is the Northeast.

Let we find the angle that vector R does with x axis.

tanα = 132/87

tanα = 1,52

α = tan-1(1,52)

α = 56,7°

The vector R is doing angle of 56.7 degrees with x axis from left side. The angle that vector R does with x axis from right side is,